
\chapter{Introduction}

The Poisson-Boltzmann Analytical and Semi-Analytical models solve the linearized Poisson-Boltzmann Equation 
(PBE) for systems hitherto not possible using traditional PBE solvers. This manual describes 
the methods and their associated suite of programs. We will refer to each method by their acronyms:
PB-AM and PB-SAM. There is much shared capability between the two softwares, so when an application
or usage can be applied to both, we will refer to both of them as PB-[S]AM, with the S in square brackets.
The PBE software suite is licensed as a collection of freely available program under a GPL license. 


\section{PB-AM} The first general analytical solution for computing the screened electrostatic 
interaction between large numbers of macromolecules of arbitrarily complex charge distributions, 
assuming they are well described by spherical low dielectric cavities in a higher dielectric 
medium in the presence of a Debye-H{\"u}ckel treatment of salt. The method exploits 
multipole expansion theory for the screened Coulomb potential such that it can describe direct 
charge-charge interactions and all higher-order cavity polarization effects between low dielectric 
spherical cavities containing their charges, while treating these higher order terms correctly 
at all separation distances. The analytical solution is general to arbitrary numbers of 
macromolecules, is efficient to compute, provides for the first time the ability to provide new 
benchmarks for other numerical solutions to the linearized Poisson-Boltzmann equation. A 
number of utilities are described below that use PB-AM results.

\section{PB-SAM} 

A semi-analytical extension of the PB-AM method. It too is capable of computing the interactions
of multiple molecules with complex charge distributions, but because of its semi-analytical approach, 
molecules are now represented as a collection of overlapping spheres. This allows for a more complex
description of the molecule in comparison to the analytical method. The applications below also apply to the 
semi-analytical method.

\section{Physical Calculations} The PB-[S]AM results allow for fast calculation of physically important quantities. 
Specifically this program can calculate the interaction energy of, and the force and torque 
applied to each molecule. These results may be written into a file or printed to the terminal.

\section{Brownian Dynamics} This package also implements dynamics simulations using the 
Brownian protocol developed by Ermak and McCammon\footnote{Ermak, D. L.; McCammon, J. A. \textit{J. Chem. Phys.} 
1978, \textit{69}, 1352\---1360.}. At each time step, the translation and rotation due to the applied 
force and torque, respectively, are calculated and added to random components of motion. The user can specify 
spatial and temporal termination conditions for the simulation and write the trajectory to specified files.

\section{Electrostatics} Another possible output of PB-[S]AM, the user can specify a configuration of an 
arbitrary number of molecules and get a 2-dimensional or 3 dimensional potential landscape of the 
system. We have provided example plotting tools, and the 3-dimensional output may be 
uploaded in VMD for visualization.

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